\documentclass[12pt]{article}
    
\renewcommand{\baselinestretch}{1.62}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{amsmath}
\usepackage[psamsfonts]{amssymb}
\usepackage{pifont}
\usepackage{enumerate}
\usepackage{theorem}
\begin{document}

\subsection{Scenario 2: Subsidy on Renewable Energy}

Introducing a subsidy on $j$ during Regime 2, the budget constraint in that
regime becomes becomes:

\begin{equation}
c+i+j(1-\tau _{j})+n+g(S,N)R+pB=y-T
\end{equation}

The budget constraint of the government is:

\begin{equation}
\tau _{j}j=T
\end{equation}

The budget constraint in Regimes 1 and 3 is the same as before, so the
analysis remains unchanged. For Regime 2, define the current value
Hamiltonian by
\begin{equation}
\begin{split}
\mathcal{H}=\frac{c^{1-\gamma }}{1-\gamma }+\lambda \biggl\{
Ak-c-i-j(1-\tau _{j})-n-g(S,N)R-p_{0}(1+H)^{-\alpha }B\biggr\}+\epsilon
(R+B-Ak)\\
+q(i-\delta k)+\eta j(1+\psi B)+\sigma QR+\nu n+\mu j+\omega n+\xi R+\zeta
B+\chi \lbrack (\bar{p}/p_{0})^{-1/\alpha }-1-H]
\end{split}
\end{equation}

The first order conditions for a maximum with respect to the control
variables are: 
\begin{equation*}
\frac{\partial \mathcal{H}}{\partial c}=c^{-\gamma }-\lambda =0
\end{equation*}%
\begin{equation*}
\frac{\partial \mathcal{H}}{\partial i}=-\lambda +q=0
\end{equation*}

\begin{equation*}
\frac{\partial \mathcal{H}}{\partial j}=-\lambda (1-\tau _{j})+\eta (1+\psi
B)+\mu =0;\mu j=0,\mu \geq 0,j\geq 0
\end{equation*}%
\begin{equation*}
\frac{\partial \mathcal{H}}{\partial n}=-\lambda +\nu +\omega =0,\omega
n=0,\omega \geq 0,n\geq 0
\end{equation*}%
\begin{equation*}
\frac{\partial \mathcal{H}}{\partial R}=-\lambda g(S,N)+\epsilon +\sigma
Q+\xi =0,\xi R=0,\xi \geq 0,R\geq 0
\end{equation*}%
\begin{equation*}
\frac{\partial \mathcal{H}}{\partial B}=-\lambda p_{0}(1+H)^{-\alpha
}+\epsilon +\eta \psi j+\zeta =0,\zeta B=0,\zeta \geq 0,B\geq 0
\end{equation*}

The differential equations for the co-state variables remain as before. Like
before, in regime 2 we have $B=Ak>0$ and $j>0$. The question here is how
effective the subsidy will be in bringing forward the time of the transition
to the renewable energy regime. Here, $q=\lambda =\frac{\eta (1+\psi Ak)}{%
(1-\tau _{j})}$, so the shadow price of energy becomes 
\begin{equation*}
\epsilon =\lambda p_{0}(1+H)^{-\alpha }-\frac{\lambda \psi j(1-\tau _{j})}{%
1+\psi Ak}
\end{equation*}%
Noting that $q=\lambda $ implies $\dot{q}=\dot{\lambda}$, we obtain 
\begin{equation*}
\frac{\dot{\lambda}}{\lambda }=\beta +\delta -A(1-p_{0}(1+H)^{-\alpha })-%
\frac{\psi Aj(1-\tau _{j})}{1+\psi Ak}
\end{equation*}
Using $\lambda =\frac{\eta (1+\psi Ak)}{(1-\tau _{j})}$ and $B=Ak$, and
using $\dot{k}=i-\delta k$, we obtain 
\begin{equation*}
\frac{\dot{\lambda}}{\lambda }=\beta -\frac{\alpha p_{0}(1+H)^{-\alpha
-1}Ak(1+\psi Ak)}{(1-\tau _{j})}-\frac{\psi A\delta k}{1+\psi Ak}+\frac{\psi
Ai}{1+\psi Ak}
\end{equation*}%
Equating the two expressions, we obtain an expression giving total
investment in regime 2, as a function of $k$ and $H$ 
\begin{equation*}
\psi A\left[ i+j(1-\tau _{j})\right] =\delta (1+2\psi Ak)-A(1+\psi
Ak)(1-p_{0}(1+H)^{-\alpha })+\frac{\alpha p_{0}(1+H)^{-\alpha -1}Ak(1+\psi
Ak)^{2}}{(1-\tau _{j})}
\end{equation*}%
The budget constraint and the first order condition for $c$ then provide a
second equation: 
\begin{equation*}
i+j(1-\tau _{j})=Ak(1-p_{0}(1+H)^{-\alpha })-\lambda ^{-1/\gamma }
\end{equation*}%
Substituting \eqref{eq:budgetreg2} into \eqref{eq:Invreg2}, we obtain an
equation relating $H$ and $k$: 
\begin{equation*}
(1+2\psi Ak)[\delta -A(1-p_{0}(1+H)^{-\alpha })]+\frac{\alpha
p_{0}(1+H)^{-\alpha -1}Ak(1+\psi Ak)^{2}}{(1-\tau _{j})}+\psi A\lambda
^{-1/\gamma }=0
\end{equation*}%
Like before, when there is active investment in both $k$ and $H$, the
investment has to maintain a relationship between the two types of capital
stocks.

Differentiating with respect to $t$, substituting $\dot{k}=i-\delta k$, $%
\dot{H}=j(1+\psi Ak)$ and for $\dot{\lambda}/\lambda $, we again obtain a
second relationship between $i$ and $j$ and the current values of $k,H$ and $%
\lambda $:$\ \ $%
\begin{equation*}
\begin{split}
& [\delta -A(1-p_{0}(1+H)^{-\alpha })]2\psi A(i-\delta k)-(1+2\psi Ak)\alpha
Ap_{0}(1+H)^{-\alpha -1})j(1+\psi Ak) \\
& +Ak(1+\psi Ak)^{2}\frac{\alpha p_{0}}{(1-\tau _{j})}(-\alpha
-1)(1+H)^{-\alpha -2}j(1+\psi Ak) \\
& +\frac{\alpha p_{0}}{(1-\tau _{j})}(1+H)^{-\alpha -1}\left[ A(1+\psi
Ak)^{2}(i-\delta k)+2A^{2}\psi k(1+\psi Ak)(i-\delta k)\right]  \\
& -\frac{\psi A}{\gamma }\lambda ^{-1/\gamma }\left[ \beta +\delta
-A(1-p_{0}(1+H)^{-\alpha })-\frac{\psi Aj(1-\tau _{j})}{1+\psi Ak}\right] 
\end{split}%
\end{equation*}



\end{document}}
